| L(s) = 1 | − 2-s + 3.31·3-s + 4-s − 0.889·5-s − 3.31·6-s − 8-s + 7.96·9-s + 0.889·10-s + 5.91·11-s + 3.31·12-s + 5.53·13-s − 2.94·15-s + 16-s − 2.38·17-s − 7.96·18-s + 4.31·19-s − 0.889·20-s − 5.91·22-s − 1.04·23-s − 3.31·24-s − 4.20·25-s − 5.53·26-s + 16.4·27-s + 29-s + 2.94·30-s − 3.16·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.91·3-s + 0.5·4-s − 0.397·5-s − 1.35·6-s − 0.353·8-s + 2.65·9-s + 0.281·10-s + 1.78·11-s + 0.955·12-s + 1.53·13-s − 0.760·15-s + 0.250·16-s − 0.579·17-s − 1.87·18-s + 0.988·19-s − 0.198·20-s − 1.26·22-s − 0.217·23-s − 0.675·24-s − 0.841·25-s − 1.08·26-s + 3.16·27-s + 0.185·29-s + 0.537·30-s − 0.568·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.121959442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.121959442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 + 0.889T + 5T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 - 5.53T + 13T^{2} \) |
| 17 | \( 1 + 2.38T + 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 + 5.78T + 37T^{2} \) |
| 41 | \( 1 + 1.25T + 41T^{2} \) |
| 43 | \( 1 + 9.99T + 43T^{2} \) |
| 47 | \( 1 + 3.52T + 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 + 0.246T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 4.68T + 67T^{2} \) |
| 71 | \( 1 + 6.98T + 71T^{2} \) |
| 73 | \( 1 + 4.99T + 73T^{2} \) |
| 79 | \( 1 + 5.82T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 4.04T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.693475280019044187435230799491, −8.375544766105621695636392216090, −7.45061425188354610630699925523, −6.89737154924313336338212376925, −6.07278476359915524856962488430, −4.43851080935509211019268171849, −3.61072121402039414206099556622, −3.29966520510094701361785031639, −1.87900549524837581307125573820, −1.30303090678075331903041804843,
1.30303090678075331903041804843, 1.87900549524837581307125573820, 3.29966520510094701361785031639, 3.61072121402039414206099556622, 4.43851080935509211019268171849, 6.07278476359915524856962488430, 6.89737154924313336338212376925, 7.45061425188354610630699925523, 8.375544766105621695636392216090, 8.693475280019044187435230799491
